Find \(\int {\arcsin x\,{\text{d}}x} \)

attempt at integration by parts with \(u = \arcsin x\) and \(v' = 1\)     M1

\(\int {\arcsin x\,{\text{d}}x}  = x\arcsin x - \int {\frac{x}{{\sqrt {1 - {x^2}} }}{\text{d}}x} {\text{ }}\)    A1A1

Note:     Award A1 for \(x\arcsin x\) and A1 for \( - \int {\frac{x}{{\sqrt {1 - {x^2}} }}{\text{d}}x} \).

solving \(\int {\frac{x}{{\sqrt {1 - {x^2}} }}{\text{d}}x} \) by substitution with \(u = 1 - {x^2}\) or inspection     (M1)

\(\int {\arcsin x{\text{d}}x} = x\arcsin x + \sqrt {1 - {x^2}} + c\)   A1

[5 marks]